y = -1/5x y = 1/2x The solution to the given system of equations is \(\mathrm{(x, y)}\). What is the...

1. TRANSLATE the problem information

• Given information:
  • \(\mathrm{y = -\frac{1}{5}x}\) (first equation)
  • \(\mathrm{y = \frac{1}{2}x}\) (second equation)
  • Need to find the value of x

2. INFER the solution strategy

• Since both equations are solved for y, we can use substitution
• Key insight: Set the two expressions for y equal to each other
• This eliminates y and gives us an equation with only x

3. SIMPLIFY by setting the equations equal

• Set \(\mathrm{-\frac{1}{5}x = \frac{1}{2}x}\)
• Move all terms with x to one side: \(\mathrm{-\frac{1}{5}x - \frac{1}{2}x = 0}\)

4. SIMPLIFY by combining fractions

• Find common denominator: LCD of 5 and 2 is 10
• Convert fractions: \(\mathrm{-\frac{2}{10}x - \frac{5}{10}x = 0}\)
• Combine: \(\mathrm{-\frac{7}{10}x = 0}\)

5. SIMPLIFY to solve for x

• When we have \(\mathrm{-\frac{7}{10}x = 0}\), this means \(\mathrm{x = 0}\)
• (Any number times 0 equals 0)

Answer: C. 0

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize that they can set the two expressions for y equal to each other. Instead, they might try to solve each equation individually, not understanding that this is a system requiring both equations to be satisfied simultaneously.

This leads to confusion because each individual equation has infinitely many solutions, and students get stuck not knowing how to find a specific x-value. This causes them to abandon systematic solution and guess.

Second Most Common Error:

Poor SIMPLIFY execution: Students make arithmetic errors when working with negative fractions, particularly when finding common denominators or combining terms. A common mistake is getting the signs wrong: writing \(\mathrm{-\frac{1}{5}x + \frac{1}{2}x = 0}\) instead of \(\mathrm{-\frac{1}{5}x - \frac{1}{2}x = 0}\).

This type of error could lead them to get a non-zero answer and select Choice D (2) or another incorrect option.

The Bottom Line:

This problem tests whether students understand that a system of equations requires finding values that satisfy ALL equations simultaneously, and whether they can execute fraction arithmetic accurately under that constraint.